Classical Morse theory revisited I -- Backward $\lambda$-Lemma and homotopy type

TL;DR

This paper introduces new tools called dynamical thickening and flow selectors to address discontinuities in gradient flows near critical points, providing a simplified proof of Milnor's homotopical cell attachment theorem within Morse theory.

Contribution

It presents the concepts of dynamical thickening and flow selectors to improve understanding of stable fibrations and offers a new, simpler proof of a key theorem in classical Morse theory.

Findings

Reproved Milnor's homotopical cell attachment theorem using new tools

Provided a conceptual framework for stable fibrations as dynamical thickenings

Enhanced understanding of gradient flow behavior near critical points

Abstract

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.